Electric field lines

What are electric field lines?

In plain words: start at any point, take a tiny step in the direction of the field there, then another step in the direction of the field at the new point, and so on; the curve you trace is a field line. It is not, in general, the path a released charge follows (see Common mistakes). Where the lines bunch together the field is strong; where they spread out it is weak.

Drawing field lines

• From arrows to lines. Around a point charge, draw arrows along the field with lengths proportional to its strength: they shrink as 1/r² and point radially outwards. Each arrow shows the force on a unit positive charge at its tail. Joining arrows that point in one direction gives a field line (NCERT page 19).
• Strength from density. Once the arrows become lines, the strength is no longer in an arrow’s length but in the density of lines: close together where the field is strong, far apart where it is weak (page 19).
• The number is a choice. Another person may draw more lines; any number can be drawn. Only the relative density in different regions matters (pages 19–20).
• Space curves. Field lines live in three dimensions, even when a figure shows them in one plane (page 21).
• Faraday’s idea. Faraday invented the picture to visualise fields without mathematics and called them lines of force. NCERT prefers field lines, since “lines of force” is somewhat misleading, especially for magnetic fields (page 20).

Density of lines and field strength

In three dimensions, the density of lines means the number of lines per unit area, counted on an area held perpendicular to them (NCERT page 20). For a point charge, NCERT shows that this density falls as 1/r², just as the field does:

• A small area ΔS held perpendicular to the lines at a distance r subtends a solid angle, NCERT’s generalisation of angle to three dimensions (page 20), measured in steradians:

• Within one solid angle the number of radial lines, n, is fixed. At distance r₁ they cross an area r₁²ΔΩ; at r₂ the same n lines cross r₂²ΔΩ. So the lines per unit area are n/(r₁²ΔΩ) and n/(r₂²ΔΩ): a 1/r² dependence (page 20).
• Put the other way: the field falls as 1/r² while a sphere around the charge grows as r², so the number of lines crossing any sphere around the charge is the same, however far away it is (page 20).
• Lines and charge. That number is density × area ∝ (q/r²) × r² ∝ q. So in a picture drawn to scale, the number of lines leaving a positive charge, or ending on a negative one, is proportional to the size of the charge: 2q has twice as many lines as q (reasoned from page 20; this is the idea behind Gauss’s law, NCERT Section 1.13).

Field lines start on positive and end on negative charges

The first general property of field lines (NCERT page 21):

• Field lines start from positive charges and end at negative charges. With a single charge, they may start or end at infinity.
• The lines of a single positive charge point radially outwards; those of a single negative charge point radially inwards.
• Around two equal positive charges, the lines bend away from the gap between them: a picture of their repulsion. Around equal and opposite charges (a dipole), lines run from +q to −q: a picture of their attraction (page 21).
• In a region with no charge, field lines can be taken to be continuous curves without any breaks (page 21).
• Where the field is the same everywhere, the lines are uniformly spaced parallel straight lines (NCERT Summary, page 38).

Field lines never cross

Two field lines can never cross each other.

NCERT Physics Part I (Class 12), Section 1.8, page 21

The reason: if two lines crossed, the field at the crossing point would have two directions, one along each tangent. But the net field at a point has a single direction, so this is impossible (page 21).

The argument needs a field that is not zero. At a null point, where E = 0 (such as the midpoint of the two equal charges in NCERT Figure 1.14(c)), the field has no direction. Limiting branches may approach the null from different directions, but they do not define a single field line passing through it with two tangents (reasoned).

One more property: electrostatic field lines do not form closed loops. This follows from the conservative nature of the electrostatic field, which NCERT takes up in Chapter 2 (page 21).

Solved examples

Common mistakes to avoid

Quick revision summary

• An electric field line is a curve whose tangent at each point gives the direction of the net field there; an arrow shows the sense.
• The relative density of lines (number per unit area, perpendicular to the lines) shows the relative strength of the field; the number drawn is a matter of choice.
• For a point charge, ΔΩ = ΔS/r^²: the same lines cross areas that grow as r^², so their density, like E, falls as 1/r^².
• Lines start on positive charges and end on negative charges (or at infinity); they are continuous in a charge-free region.
• Two field lines never cross, and electrostatic field lines form no closed loops; a uniform field has parallel, equally spaced lines.

Questions students ask

Why do electrostatic field lines not form closed loops?

Electrostatic field lines cannot form closed loops because the electrostatic field is conservative, as NCERT notes on page 21: the work it does on a charge moved from one point to another does not depend on the path, so its work round any closed path is zero (NCERT Chapter 2, pages 45–46). If a field line closed on itself, the field would push a positive charge carried once round it forwards all the way, doing positive work round a closed path, which is impossible. So the lines must start on positive charges and end on negative charges or at infinity; the rule is for fields of charges at rest.

What is the difference between electric field lines and magnetic field lines?

Electrostatic field lines start on positive charges and end on negative charges, or at infinity, and never form closed loops, while the magnetic field lines of a magnet or a solenoid form continuous closed loops (NCERT Chapter 5, page 137). The difference comes from the sources: isolated positive and negative charges exist, but fundamental magnetic monopoles have not been established experimentally. In the standard no-monopole description, magnetic field lines have nowhere to begin or end (NCERT page 151). Both kinds of line give the field’s direction by their tangent, crowd where the field is strong and never cross (pages 19–21 and 138), but, unlike electric field lines, magnetic ones do not show the direction of the force on a moving charge (page 137).

Are electric field lines real?

Electric field lines are not real objects but a picture: Faraday invented them as an intuitive, non-mathematical way of visualising the field, and NCERT calls them a way of mapping the field around a configuration of charges (NCERT page 20). The field they map is real, though: NCERT regards electric and magnetic fields as physical entities that carry energy, not mere mathematical constructs (page 17).

What is the difference between electric field and electric field lines?

The electric field is a vector quantity with a definite size and direction at every point in space, the force per unit positive test charge there, in N C^⁻¹, while electric field lines are curves drawn to picture it (NCERT pages 14–16 and 20). A field line shows the field’s direction by its tangent and, through how closely it crowds its neighbours, the field’s relative strength, but it does not give the field’s value at a point (NCERT pages 20–21).

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. What does the tangent to a field line at a point give?

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   The direction of the net electric field at that point (NCERT pages 20–21).

2. Q2. Around a point charge, the lines per unit area at distance r are N. What are they at 2r?

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   N/4, because the same lines cross an area 4 times as large (NCERT page 20).

3. Q3. Who invented the picture of field lines, and what did he call them?

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   Faraday, who called them lines of force (NCERT page 20).